Two curious congruences for the sum of divisors function
aa r X i v : . [ m a t h . N T ] F e b TWO CURIOUS CONGRUENCES FOR THE SUM OFDIVISORS FUNCTION
HARTOSH SINGH BAL AND GAURAV BHATNAGAR
Abstract.
We show two congruences involving the sum of the divisorsfunction σ ( n ). The results follow from recurrence relations for the sumof divisors function. Introduction
For a given positive integer n , let σ ( n ) denote the sum of its positivedivisors. We take σ ( n ) = 0 if n is not a positive integer. The objective ofthis note is to prove the following congruences. Theorem 1.1.
Let n be a positive integer. If ∤ n , ∞ X j =0 σ (cid:0) n + 1 − j ( j + 1) (cid:1) ≡ . (1.1) If n = k ( k + 1) / for some k , then ∞ X j =0 σ (cid:0) n − j ( j + 1) / (cid:1) ≡ . (1.2)Congruences such as 3 | σ (3 n + 2) and 4 | σ (4 n + 3) easily follow from thedefinition of σ ( n ) by considering possible factorizations of 3 m + 2 and 4 n + 3.While Bonciocat [2] proved congruences for the convolution of the sum ofdivisors functions (also see Gallardo [4]), no congruences of a similar natureseem to be known.2. Recurrences for the Sum of Divisors Function
We begin by proving three recurrence relations for the sum of divisorsfunction.
Theorem 2.1.
Let n be a positive integer. The following recursions holdfor σ ( n ) : nσ (2 n + 1) = ∞ X j =1 (cid:0) j ( j + 1) − n (cid:1) σ (cid:0) n + 1 − j ( j + 1) (cid:1) ; (2.1) Date : February 23, 2021.2010
Mathematics Subject Classification.
Primary: 11A07 Secondary: 11A25.
Key words and phrases. sum of divisors function, congruences, recurrence relations,sums of triangular numbers. ∞ X j =0 (cid:16) σ ( n − j ( j + 1) / − σ (cid:0)
12 ( n − j ( j + 1) / (cid:1)(cid:17) = ( n, if n = j ( j + 1) / for some j ;0 , otherwise ; (2.2) nσ (2 n + 1) = 4 ∞ X j =1 (cid:0) σ ( j ) − σ ( j/ (cid:1) σ (2 n + 1 − j ); (2.3) where we take σ ( m/
2) = 0 if m is odd.Proof. In [1] we considered powers of the theta function ψ ( q ) = ∞ X k =0 q k ( k +1) / = ∞ Y k =1 (1 − q k )(1 − q k +1 ) − to obtain the recursion [1, Eq. (6.6b)] nt k ( n ) + ∞ X j =1 (cid:0) n − ( k + 1) j ( j + 1) / (cid:1) t k ( n − j ( j + 1) /
2) = 0 (2.4)where t k ( n ) the number of ways of writing n as an ordered sum of k triangu-lar numbers. Here, we take t k (0) = 1 and t k ( m ) = 0 for m <
0. Now recalla result due to Legendre (attributed by Ono, Robins and Wahl [6, Th. 3]) t ( n ) = σ (2 n + 1) . Substituting this in (2.4), we obtain (2.1).To prove (2.2) and (2.3) we again consider ψ ( q ). By [1, Eq. (3.3b)] we obtain (cid:16) ∞ X n =0 q n ( n +1) / (cid:17) ∞ X k =1 g ( k ) q k = ∞ X j =0 j ( j + 1) q j ( j +1) / (2.5)where g ( n ) is given by g ( n ) = ( σ (2 m − , for n = 2 m − , m = 1 , , . . . ; (cid:0) σ o (2 m ) − σ e (2 m ) , for n = 2 m, m = 1 , , . . . . Here σ o ( n ) (respectively, σ e ( n )) denotes sum of odd divisors (respectively,even divisors) of n . From σ o (2 m ) = σ (2 m ) − σ e (2 m ) , and, σ e (2 m ) = 2 σ ( m ) , we obtain g ( m ) = σ ( m ) − σ ( m/ , where we take σ ( m/
2) = 0, if m is odd. WO CURIOUS CONGRUENCES FOR THE SUM OF DIVISORS FUNCTION 3
Substituting this expression for g ( n ) in 2.5, we compare coefficients of q k for k >
0, to obtain (2.2).We then consider the coefficients of the fourth power of ψ ( q ) to obtainanother expression for t ( n ) by [1, Eq. (3.3b)], nt ( n ) = 4 ∞ X j =1 g (cid:0) j (cid:1) t (cid:0) n − j (cid:1) . (2.6)Substituting t ( n ) = σ (2 n + 1) in this expression we get (2.3). (cid:3) Similar recurrences have been proved by Ewell [3] but they do not yieldcongruences of the kind listed here. The sequence g ( n ) = σ ( m ) − σ ( m/ Proof of Theorem 1.1.
If 5 ∤ n , we can take the identity (2.1) mod 5 andcancel the n to obtain obtain (1.1). If n = j ( j + 1) / j , then wecan take the identity (2.2) mod 4 to obtain (1.2). (cid:3) If we consider identity (2.3) mod 4 we trivially obtain nσ (2 n + 1) ≡ n to be odd, i.e., of the form 2 m + 1 for m a positiveinteger, we also recover σ (4 m + 3) ≡ References [1] H. S. Bal and G. Bhatnagar. The partition-frequency enumeration matrix, 2021.arXiv:2102.04191.[2] N. C. Bonciocat. Congruences for the convolution of divisor sum function.
Bull. GreekMath. Soc. , 47:19–29, 2003.[3] J. A. Ewell. Recurrences for the sum of divisors.
Proc. Amer. Math. Soc. , 64(2):214–218, 1977.[4] L. H. Gallardo. On Bonciocat’s congruences involving the sum of divisors function.
Bull. Greek Math. Soc. , 53:69–70, 2007.[5] H. Movasati and Y. Nikdelan. Gauss-Manin connection in disguise: Dwork family,2017. arXiv:1603.09411.[6] K. Ono, S. Robins, and P. T. Wahl. On the representation of integers as sums oftriangular numbers.
Aequationes Math. , 50(1-2):73–94, 1995.[7] N. J. A. Sloane. The on-line encyclopedia of integer sequences. https://oeis.org ,2003.
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